Integrand size = 35, antiderivative size = 212 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=-\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {a c^2 \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)} \]
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Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1015, 847, 794, 201, 223, 212} \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=-\frac {a c^2 \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]
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Rule 201
Rule 212
Rule 223
Rule 794
Rule 847
Rule 1015
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x} \\ & = \frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (-4 b^2 c+10 a b d x\right ) \sqrt {c+d x^2} \, dx}{5 d \left (2 a b+2 b^2 x\right )} \\ & = \frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+d x^2} \, dx}{2 d \left (2 a b+2 b^2 x\right )} \\ & = -\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )} \\ & = -\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{4 d \left (2 a b+2 b^2 x\right )} \\ & = -\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {a c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.50 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (\sqrt {c+d x^2} \left (15 a d x \left (c+2 d x^2\right )+8 b \left (-2 c^2+c d x^2+3 d^2 x^4\right )\right )+15 a c^2 \sqrt {d} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{120 d^2 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.49
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (24 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{\frac {3}{2}} b \,x^{2}+30 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{\frac {3}{2}} a x -16 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {d}\, b c -15 \sqrt {d \,x^{2}+c}\, d^{\frac {3}{2}} a c x -15 \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) a \,c^{2} d \right )}{120 d^{\frac {5}{2}}}\) | \(103\) |
risch | \(\frac {\left (24 b \,x^{4} d^{2}+30 a \,x^{3} d^{2}+8 b c \,x^{2} d +15 a c x d -16 b \,c^{2}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {\left (b x +a \right )^{2}}}{120 d^{2} \left (b x +a \right )}-\frac {c^{2} a \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) \sqrt {\left (b x +a \right )^{2}}}{8 d^{\frac {3}{2}} \left (b x +a \right )}\) | \(112\) |
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Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.83 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\left [\frac {15 \, a c^{2} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt {d x^{2} + c}}{240 \, d^{2}}, \frac {15 \, a c^{2} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt {d x^{2} + c}}{120 \, d^{2}}\right ] \]
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\[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\int x^{2} \sqrt {c + d x^{2}} \sqrt {\left (a + b x\right )^{2}}\, dx \]
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\[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\int { \sqrt {d x^{2} + c} \sqrt {{\left (b x + a\right )}^{2}} x^{2} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.55 \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\frac {a c^{2} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{8 \, d^{\frac {3}{2}}} + \frac {1}{120} \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (3 \, {\left (4 \, b x \mathrm {sgn}\left (b x + a\right ) + 5 \, a \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {4 \, b c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x + \frac {15 \, a c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x - \frac {16 \, b c^{2} \mathrm {sgn}\left (b x + a\right )}{d^{2}}\right )} \]
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Timed out. \[ \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx=\int x^2\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+c} \,d x \]
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